100 Prof. Oayley. On the [June 19, 



them to pass through one and the same point of the surface ; we have 

 then the geodesies drawn in all directions through this point of the 

 surface; doing the same thing with the projections, we have, it is 

 clear, the projections of the geodesies drawn in all directions through 

 the point. It is easy, by drawing the projections each on a separate 

 circle of paper, and passing a pin through the centres, to form a 

 model by means of which an accurate figure of the projection may be 

 constructed. But I content myself with a mere diagram (fig. 4). 



FIG. 4. 



26. Taking a point Q so low down on the surface that the 

 geodesic at right angles to the meridian through Q is a simple arc 

 A' A, then imagining the two extremities A, A' each moving in 

 the same sense round the circle, bat A faster than A', so as to assume 

 the positions B, B' ; C, C' ; and so on to K, K' coinciding with each 

 other, we have the arcs B'B, C'C, and so on until we come to the loop 

 form K'K : after which we have I/ in advance of L, and so on to 

 curves of any number of convolutions. Considering any two arcs 

 B'B, C'C and drawing the geodesic BC which joins their extre- 

 mities B and C. then any geodesic through Q intermediate to B'B, 

 C'C, or, say, to QB, QC, will meet the arc BC ; while the geodesies 

 through Q extramediate to QB, QC will not meet, or will only after a 

 Convolution or convolutions meet, the arc BC. This of course corre- 

 sponds to the Lobatschewskian theory, according to which we have 

 through a point Q to the extremities at infinity of a line BC, two 

 distinct lines QB, QC, said to be the parallels through Q of the line 

 BC; and which are such that any line through Q intermediate to 



